It’s pretty simple. In 1 Samuel, we see that after David is anointed King in Saul’s stead, he is invited to be Saul’s harp player. God sent an evil spirit to torment Saul because he had departed from God and had gone his own way.
So whenever the evil spirit tormented him, David played the harp in his court and Saul was soothed.
Then one day Goliath threatened Israel. David came forward (because at this point the Holy Spirit was upon him, he had been anointed king) and said I’ll kill him.
He went to Saul and told him. Saul gave David his armor and sent him on his way. David before leaving, declined the armor, because he had not proved the armor. Saul said, ok, go. Godspeed.
So Saul knew David.
Now after Goliath was killed by David, Saul said, who is this one who killed Goliath? Bring him to me. So David stood before Saul and Saul said who are you? David said I am Jesse’s son.
Now why did Saul treat David like he didn’t know him? I don’t know. But there are many possibilities. Easy possibilities.
1-Saul was at this point tormented by an evil spirit so he could easily be out of his mind and not making sense.
2-David was playing the harp in Sauls court when Saul was tormented so we don’t know how incoherent Saul was during these episodes. He often tried to kill David as he was playing the harp so Saul could have been totally possessed or totally oppressed. I don’t know if Saul was a believer. That is kind of left open. He fell away into disobedience. We do know that.
3-It’s entirely possible that Saul paid no attention whatsoever to who was playing the harp.
4-When David came to Saul to ask him if he could go kill Goliath, it’s possible that Saul paid no attention to the boy, or to what he looked like or who he was. That is very possible. We aren’t given any details but the bare minimum. It could have been a huge loud commotion with many distractions. I can easily see that.
Anyway, I can see a lot more possibilities. So why didn’t Saul recognize David? Many many possibilities.
God bless you as you study His word!